Module 8: Benefit-Cost Analysis and Dynamic Decision-Making

So far in this class we have talked about:

In this module, we are going to look at how to compare between benefits and costs. We will talk about:

One of the most important reasons to environmental benefits and costs is to evaluate public policy decisions:

And these projects need to be funded by the public

The Decision Rule

Let B be the benefits from a proposed policy, and C be the costs. The decision rule is:

If B>C, do it. Otherwise, don’t do it.

Alternatively,

If the benefit-cost ratio B/C>1, do it.

Benefit-cost Analysis

Benefit-cost analysis provides a normative criteria to evaluate public policy decisions

Of course, benefit-cost analysis is NOT the only decision rule:

And there is this

Executive Order 13771 (1/30/2017): “(c) …any new incremental costs associated with new regulations shall, to the extent permitted by law, be offset by the elimination of existing costs associated with at least two prior regulations.”

And benefit-cost analysis does not guarantee economic efficiency

Mike Pompeo on the Paris Agreement, 11/4/2019

https://finance.yahoo.com/video/mike-pompeo-explains-why-us-002029088.html

Where does the Secretary of State get wrong?

What does benefit-cost analysis look like?

An inherent puzzle with public investments

Cleaning up a waste site near Kingslanding costs $25 million in year 2019. Starting from 2020, it provides economic benefits of $10 million per year for the next three years, before another cleanup is due.

How to evaluate the benefits and the costs?

Discounting the future

In general, we might want to value more heavily with money on hand. This is true for several reasons:

This leads to a positive discount rate of r

Present value of benefits & costs

With the discount rate established, we can now evaluate the benefits and costs for the waste cleanup problem.

Assuming the base year is 2019, and a discount rate of 5%. Our problem is to put every benefit stream occuring in the future back into 2019 terms. For example, a benefit of $10 million in 2020 is worth:

$10 million in 2020 = $10 million / (1+5%) = $9.52 million in 2019
$10 million in 2021 = $10 million / (1+5%)^2 = $9.07 million in 2019
$10 million in 2022 = $10 million / (1+5%)^3 = $8.64 million in 2019

The present value benefits and costs for the project is:

The present value benefits: \[PVB = 10 * \frac{1}{(1+5\%)}+10 * \frac{1}{(1+5\%)^2} + 10 * \frac{1}{(1+5\%)^3}\]

= 9.52+9.07+8.64
= $27.23 million

The present value costs: PVC = $25 million

Present value net benefits

The present value net benefits is thus:

PVNB = PVB - PVC
=27.23-25 = $2.23 million

Present value vs. current value

And we have the following formula:

\[PV = \frac{CV}{(1+r)^t}\]
\[CV = PV * (1+r)^t\]

where
PV: Present value of benefits
CV: Current value of benefits
r: Discount rate
t: Number of periods

Present value benefits for the infinite horizon

In many circumstances, future benefit/cost streams will run for a long period of time, or indefinitely:

And how do we put that into account?

The construction of the Three Gorges Dam cost $30 billion in year 0. Starting from year 1, it provides economic benefits of $1 billion per year until the earth is occupied by the three-body civilization. Assume a discount rate of 5%.

Current value costs: $30 billion in year 0
Current value benefits: $1 billion starting in year 1, indefinitely

Present value costs: $30 billion
Present value benefits:

\[ 1 / 1.05 + 1/1.05^2 + 1/1.05^3 + 1/1.05^4 + ...\]

using the summation rule of geometric sequences: \[\sum_{s=1}^t A * \frac{1}{(1+r)^s} = A * \frac{1}{1+r}* \frac{1 - (\frac{1}{1+r})^t}{1 - \frac{1}{1+r}}\]

When t goes to infinity, we have: \[\sum_{s=1}^t A * \frac{1}{(1+r)^s} = A * \frac{1}{1+r}* \frac{1}{\frac{r}{1+r}}\]
\[= \frac{A}{r}\]

Or:

PV of an infinite stream of payoff = \(\frac{\text{CV of each year's payoff}}{r}\)

And the PVB for the project is: 1/0.05 = $20 billion.
The PVNB is thus: 20-30 = $-10 billion

The influence of discount rate

Discount rate has a critical impact on benefit-cost analysis. Suppose for the same project ($30B initial cost, $1B indefinite benefit), different discount rates are chosen:

##  Discount_rate PVB PVC  PVNB
##           0.01 100  30  70.0
##           0.02  50  30  20.0
##           0.03  33  30   3.3
##           0.05  20  30 -10.0
##           0.10  10  30 -20.0
##           0.20   5  30 -25.0

Internal Rate of Return

Another way to evaluate policy projects (or business ones) is to calculate the internal rate of return.

IRR = the discount rate when the project is just breaking even

Put it in another way

From the financial markets’ perspective:

From consumers’ perspective

There are (at least) two main reasons why an individual values present wealth more than future wealth:

Social vs. private discount rate

But the problem with a market-driven discount rate is, private discount rates are higher than the social discount rate

By the same logic, developed countries may have lower interest rate than developing countries:

This is to say, while US should use a discount rate of 3%, Mozambique should use a discount rate of 8%.

What are the implications of this?

Frank Ramsey and the Ramsey equation

Frank Ramsey is NOT an economist. He was a philosophy/mathematician working with the great Wittgenstein. He was encouraged by John Maynard Keynes and Arthur Pigou(!) though.

Ramsey wrote only three economics papers in his life:

Paul Samuelson: “three great legacies – legacies that were for the most part mere by-products of his major interest in the foundations of mathematics and knowledge.”

John Maynard Keynes: “one of the most remarkable contributions to mathematical economics ever made, both in respect of the intrinsic importance and difficulty of its subject…”

The Ramsey equation

Ramsey (1928): How to maximize present value benefits over time? Or rather, how should the society tradeoff current consumption with future ones?

After some difficult (but elegant) math, it goes as simple as this:

\[r = \delta + \eta g\]

where:
r: discount rate
\(\delta\): pure time preference
\(\eta\): elasticity of marginal utility of consumption
g: growth rate of the economy

In other words, discount rate consists of two part:

Some estimates of the social discount rate

Consequences of discount rate choices

Drastic differences on the social cost of carbon (price of a carbon tax):

Discount rate as a political tool

And of course, discount rate is a power weapon if politicians want to “play with the numbers”:

Nordhaus vs. Weitzman vs. Stern

What do you think the discount rate should be?

Decision-making under uncertainty

In the real world, it is hard to state the consequence of a policy with certainty.

How does benefit-cost analysis incorporate uncertainty?

A government can implement one of the three programs to restore a migratory bird habitat. Three plans have the same cost of $1000. The economic benefits from each strategy is the following:

Outcome A Outcome B Outcome C
Economic Benefits 500 1000 2000
Plan A 30% 50% 20%
Plan B 10% 65% 25%
Plan C 20% 40% 40%

Which plan should the government pick?

Dominant policy

A dominant policy is one that confers higher net benefits for every outcome.

Also, a dominated policy is a policy that confers lower net benefits for every outcome comparing to another policy.

Is there a dominant strategy there? Is there a dominated strategy there?

Expected value maximization

If there is no dominant strategy, then we will have to rely on something else: maximize the expected value of net benefits

Expected Net Benefit of Choice j = \(\sum_{i=1}^I p_i NB_{ij}\)

Plan A: E(NB) = 500 * 30% + 1000 * 50% + 2000 * 20% = 1050

Plan B: E(NB) = 500 * 10% + 1000 * 65% + 2000 * 25% = 1200

Plan C: E(NB) = 500 * 20% + 1000 * 40% + 2000 * 40% = 1300

Another example

Honduras is thinking of a national climate change adaptation plan that costs $70 million. There is a 70% chance that the plan is successful, and create an annual benefit of $4 million indefinitely. There is a 30% chance that the plan is unsuccessful, creating only $1 million indefinitely.

Assuming a discount rate of 5%, should the government go through with the plan?

Maximizing expected present value net benefits

E(PVNB) = 4,000,000/5% * 70% + 1,000,000/5% * 30% - 70,000,000
= -$8,000,000

Honduras should not go through with that plan.

Takeaways from the module